Is arc connected-ness well-behaved with respect to products?

That is -

$\prod X_\alpha$ is arc connected iff $X_\alpha$ is arc connected $\forall \alpha$

In this question on MathStackexchange, an answer is provided only for the reverse implication, that is -

If $X_\alpha$ is arc connected $\forall \alpha$, then $\prod X_\alpha$ is arc connected

However, I wasn't able to get an answer for the forward implication, nor have I been able to find it in any book or from Googling. So, is the forward implication true, or is there a counterexample for the same?